A126164 Sum of the proper exponential divisors of n.

0, 0, 0, 2, 0, 0, 0, 2, 3, 0, 0, 6, 0, 0, 0, 6, 0, 6, 0, 10, 0, 0, 0, 6, 5, 0, 3, 14, 0, 0, 0, 2, 0, 0, 0, 36, 0, 0, 0, 10, 0, 0, 0, 22, 15, 0, 0, 18, 7, 10, 0, 26, 0, 6, 0, 14, 0, 0, 0, 30, 0, 0, 21, 14, 0, 0, 0, 34, 0, 0, 0, 48, 0, 0, 15

Offset

1,4

Comments

The e-divisors (or exponential divisors) of x=Product p(i)^r(i) are all numbers of the form Product p(i)^s(i) where s(i) divides r(i) for all i.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Ant King, Mathematica programs for A126164 - A126166.

Eric Weisstein's World of Mathematics, e-Divisor.

Formula

a(n) = esigma(n) - n = A051377(n) - n.

Example

The exponential divisors of 240 are 30, 60 and 240, so a(240) = 30+60 = 90.

Mathematica

f[p_, e_] := DivisorSum[e, p^# &]; a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - n; Array[a, 100] (* Amiram Eldar, Aug 13 2023 *)

Prog

(PARI)
A051377(n) = { my(f=factor(n)); prod(i=1, #f[, 1], sumdiv(f[i, 2], d, f[i, 1]^d)); }; \\ This function from Charles R Greathouse IV, Nov 22 2011
A126164(n) = (A051377(n) - n); \\ Antti Karttunen, Oct 04 2017, after the given formula

Crossrefs

Cf. A051377, A049419, A054979, A054980, A126168.

Sequence in context: A349435 A063958 A349434 * A340317 A145007 A308376

Adjacent sequences: A126161 A126162 A126163 * A126165 A126166 A126167

Keyword

easy,nonn

Author

Ant King, Dec 21 2006

Status

approved